IPPR 001

Image Processing Operations

Point Operation

$b[m, n] = f(a[m, n])$

• It only depends on the value of the pixel itself, not on the values of its neighbors.
• e.g. current pixel + 20.
• to increase the brightness of an image, adjust contrast, or apply a threshold to create a binary image.

Local Operation

$b[m, n] = f(a[m - 1, n - 1], a[m - 1, n], a[m - 1, n + 1], a[m, n - 1], a[m, n], a[m, n + 1], a[m + 1, n - 1], a[m + 1, n], a[m + 1, n + 1])$

• It depends on the values of the pixel and its neighbors.
• e.g. current pixel + average of 8 neighbors.
• to blur an image, sharpen an image, detect edges, or convolution with a kernel.
• The most common type of neighborhoods are:
  • 4-neighbors:
    • top, bottom, left, right.
  • 8-neighbors:
    • top, bottom, left, right, and the 4 diagonal neighbors.

Global Operation

$b[m, n] = f(a[0, 0], a[0, 1], ..., a[M - 1, N - 1])$

• It depends on the values of all pixels in the image.
• e.g. current pixel + average of all pixels in the image.
• to compute the histogram equalization, apply a global threshold, or perform a Fourier transform.

Image Histogram

• It is a graph showing how many pixels in an image have each possible intensity value.
  • Intensity value: the brightness of a pixel.
• e.g. 8-bit grayscale image has 256 possible intensity values (0-255).
  • The histogram will graphically display 256 numbers showing the distribution of pixels among those gray-scale values.

Histogram Equalization

0                     255
|------████████--------|
       80~140에 몰림

0                     255
|--██--██--██--██--██--|

• It spreads out the intensity values that are concentrated in a narrow range, increasing the contrast of the image.
• It is useful when the images have been acquired under poor lighting conditions or have low contrast (different circumstances).

Noise

• Any undesired information that contaminatest the image.
• During the analog-to-digital conversion process, it is a side effect of the physical conversion of patterns of light energy into electrical patterns.
• The shape of distribution of noise types used to describe many of them and is related closely to the histogram.

Gaussian Noise

frequency
 ^
 |             █
 |          █████
 |       █████████
 |    █████████████
 +----------------------> noise gray level
        -20  0  +20

• The most common type of noise, with a bell-shaped distribution.
• Natural noise process such as electronic noise in the image acquisition system.

Uniform Noise

frequency
 ^
 |       ┌───────────────┐
 |       │               │
 |       │               │
 +-------┴───────────────┴------> noise intensity
         a               b

• A type of noise with a distribution that is constant across the range of intensity values.
• The gray-level values of noise are evenly distributed across a specific range.
• It can be used to generate any toehr type of noise distribution, often used to degrade images for the evaluation of image restoration algorithms.
  • it provides the most unbiased or neutral noise model.

Salt-and-pepper noise

frequency
 ^
 | █                       █
 | █                       █
 | █                       █
 +----------------------------> gray level
   0                         255

• A distribution that has two spikes at the minimum and maximum intensity values.
• The presence of single dark pixels in bright regions, or single bright pixels in dark regions.
  • Typically affects a small set of pixels.
• It is usually quantified by the percentage of pixels which are corrupted by noise.
• It is typically caused by errors in data transmission, faulty memory locations, or malfunctioning pixel elements in camera sensors.

Signal-to-Noise Ratio

$SNR = 10 \log_{10} \frac{P_{signal}}{P_{noise}}$

• SNR
• The ratio between the power of the signal and that of the noise.
• In a perfect image, the ratio of signal to noise is infinite.

Noise Elimination

• Restore the true value of the pixels as much as possibole.
• It may undesirably reduce image information.
• Averaging the pixel with its neighbours will smooth the noise or other types of image filters can be applied to reduce noise.

Filters

• Linear filters: low pass, high pass
• Non-linear filters: median
• Filters are used to improve an image
  • if the image is destined for human viewing, to make it more pleasant to look it or more readable.
  • if the image is the input to a pattern recognition process, to facilitate the following steps of automated image analysis.

Convolution

$I(r, c) \otimes F = \sum_{i=1}^{2M + 1} \sum_{j=1}^{2M+1} I(r+i-(M+1), c + j-(M+1)) F(i, j)$

• Multiply the pixels of a neighborhood of $(r, c)$ by the corresponding coefficients of the filter $F$, and add them all together.

Low Pass Filter

• Smoothing or softening, employes to remove high spatial frequency noise from a disital image.
• It replace each pixwel with a weighted sum of each pixel's neighbors.
• It is used to remove noise, might have the side-effect of generally smoothing or blurring images and reducing edge information.
• Local averaging: take the local average of the pixels in a neighborhood and replace the center pixel with that value.

Gaussian Filter

$H_{ij} = \frac{1}{2\pi\sigma^2} e^{-\frac{i^2 + j^2}{2\sigma^2}}$

• yields a $2k+1 \times 2k+1$ kernel, where $k$ is the size of the filter and $\sigma$ is the standard deviation of the Gaussian distribution.
• A smoothing filter that computes a weighted average of neighboring pixels, giving larger weights to pixels closer to the center.

1   4   7   4   1
4  16  26  16   4
7  26  41  26   7
4  16  26  16   4
1   4   7   4   1

• Smaller $\sigma$ values result in a more localized filter, which means weak smoothing and less blurring of the image.
• Larger $\sigma$ values result in a more spread-out filter, which means stronger smoothing and more blurring of the image.

Median Filter

• A non-linear filter that replaces a pixel with the median of its neighbors.
• It is effective at removing salt-and-pepper noise and other isolated noise compared to low-pass linear filters.
• Less blurred, edges remain sharp, removes single pixel erros completely, but slower requires sorting the pixels in the neighborhood.

10  11  10
12 255  11
10  12  11

# 255 is salt-and-pepper noise, the median of the 9 pixels is 11, so the center pixel is replaced with 11.
10, 10, 11, 11, 11, 12, 12, 255

# to-be
10, 10, 11, 11, 11, 12, 12, 11

High Pass Filter

• It extracts high-frequency components, such as edges and fine details, by subtracting a low-pass filtered image from the original image.
• Sometimes, it is desired to enhance the high frequencies without removing the low frequencies.

Sharpened Image = Original Image + High-frequency component
  = Origial Image + (Original Image - Low-pass filtered Image)

Conclusion

• Low-pass filter → smooth / blur
• High-pass filter → edge / detail
• High-pass + original → sharpening